Understanding jump discontinuity in disordered system

The response of a complex system to a slow varying external force often displays a jump discontinuity in the order parameter near the critical point. However, this discontinuity is not usually a single jump but rather breaks into smaller jumps which makes it difficult to locate the critical point on approaching its vicinity based only on simulations, in the absence of exact results. Our work is a small effort in understanding these breaks in jump through the hysteretic response of a classical Ising spin system to an external field, $h$, in the context of a nonequilibrium zero-temperature random field Ising model on dilute systems. We consider a Bethe lattice with coordination number, $z = 4$, and dilute a fraction $(1-c)$ of the sites. Therefore the lattice now consists of sites with varying $z = 4, 3, 2, 1$ and possibly few isolated sites $(z=0)$, depending on the concentration $c$. We obtain the exact solution of the magnetization curve, $m(h)$ vs $h$, for the entire lattice as well as for each sublattice of different $z$ coordinated sites, $m_4(h), m_3(h), m_2(h), m_1(h), m_0(h)$. The discontinuity in total magnetization is the result of the superposition of the jumps of different $z$ coordinated sites and observed at the same value of external field, $h_{crit}$. The dominant contribution to the jump comes from those sites with higher concentration and larger $z$. However, the triggering sites responsible for large jumps are mostly $z\ge3$. We test this on cubic lattices as well, where exact results are not available. We hope our analysis will help in understanding fluctuations around a jump in numerical simulations as well as experiments.